<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">pmet</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">pmet</font> <a href="nagata_1.html#R1">is_a_pseudometric_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> holds   <a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Olive">p</font> is <a href="pscomp_1.html#V9">continuous</a>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>reconsider </b><font color="Maroon">infR</font> =  <a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>reconsider </b><font color="Maroon">dR</font> =  <a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Maroon">t</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E2:13_1"/>
for <font color="Olive">R</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Olive">R</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">infR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">R</font> holds <br/> ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Maroon">infR</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Olive">R</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">R</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">R</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">infR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">infRt</font> = <font color="Maroon">infR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">t</font>, <font color="Maroon">dRt</font> = <font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">t</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:13_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">infRt</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">dB</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">dRt</font>,<font color="Maroon">r</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E6:13_1_1"/>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="nagata_2.html#K3" target="_self">dist</a> <font color="Maroon">pmet</font>,<font color="Maroon">t</font> is <a href="pscomp_1.html#V9">continuous</a> )
 
<b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>
<a NAME="E7:13_1_1"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">dRt</font>,<font color="Maroon">dRt</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">dR</font> is <a href="pre_topc.html#V5">continuous</a> )
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T15">TOPMETR:15</a>, <a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>
<a NAME="E8:13_1_1"/><b>then </b>
( <font color="Maroon">dRt</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dB</font> &amp; <font color="Maroon">dB</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">dR</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">t</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a>, <a class="ref" href="tmap_1.html#T55">TMAP_1:55</a>, <a class="ref" href="topmetr.html#T21">TOPMETR:21</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E10:13_1_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">dR</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">dB</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/>
<a NAME="E11:13_1_1"/>
<font color="Maroon">infR</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Ib</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_1"/>
<font color="Maroon">Ib</font> <a href="hidden.html#R2">in</a> <font color="Maroon">infR</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">infR</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">infR</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ib</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">infRb</font> = <font color="Maroon">infR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">b</font>, <font color="Maroon">dRb</font> = <font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E6:13_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">dR</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:13_1_1_1"/><b>then </b>
( <font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dR</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">B</font> &amp; <font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">t</font>,<font color="Maroon">t</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E8:13_1_1_1"/><b>then </b>
( <font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">dB</font> &amp; <font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="nagata_1.html#T28">NAGATA_1:28</a></i>;<br/>
<a NAME="E9:13_1_1_1"/><b>then </b>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">dRt</font>,<font color="Maroon">dRb</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">dRt</font>,<font color="Maroon">dRb</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default">0 <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> &amp; <font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">t</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a>, <a class="ref" href="nagata_1.html#T29">NAGATA_1:29</a>, <a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E10:13_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">0 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<a NAME="E11:13_1_1_1"/><b>then </b>
<font color="Maroon">dR</font> <a href="nagata_2.html#K1" target="_self">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E12:13_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">t</font>,<font color="Maroon">b</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">infRt</font>,<font color="Maroon">infRb</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span>)</span> &amp; <font color="Maroon">pmet</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">t</font>,<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, , , <a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E13:13_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">infRt</font>,<font color="Maroon">infRb</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:13_1_1_1"/><b>then </b>
<font color="Maroon">infRb</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">infRt</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E15:13_1_1_1"/>
<font color="Maroon">Ib</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:13_1_1"/>
 ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">U</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Maroon">infR</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon">infR</font> <a href="tmap_1.html#R1">is_continuous_at</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48">TMAP_1:48</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:13"/><b>then </b>
<font color="Maroon">infR</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T55">TMAP_1:55</a></i>;<br/>
<b>hence </b><a NAME="E6:13"/>
 <a href="nagata_2.html#K4" target="_self">inf</a> <font color="Maroon">pmet</font>,<font color="Maroon">A</font> is <a href="pscomp_1.html#V9">continuous</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T83">TOPREAL6:83</a></i>;<br/>


</div>
